Question

How do you solve the following system of equations: \[3x + 2y = - 7\], \[5x - 4y = 19\]

Answer 1

Hint: In this question we have to find the unknown values that are \[x\] and \[y\] solving these equations by using the elimination method. In elimination methods either we add or subtract the equations to find the unknown.

Complete step-by-step answer:
Let us consider the equation and we will name it as (1) and (2)
\[3x + 2y = - 7\]-------(1)
\[5x - 4y = 19\]-------(2)
Now we have to solve these two equations to find the unknown
Multiply (1) by 5 and (2) by 3 then we get
\[15x + 10y = - 35\]
\[15x - 12y = 57\]
Since the coordinates of \[x\] are the same and we need not to change the sign by the alternate sign. we simplify to known the unknown value \[y\]
\[
   + 15x + 10y = - 35 \\
 \underline{ \mathop + \limits_{( - )} 15x\mathop - \limits_{( + )} 12y = \mathop + \limits_{( - )} 57} \\
  \]
Now we cancel the \[x\] term so we have
\[
 + 15x + 10y = - 35 \\
\underline { \mathop + \limits_{( - )} 15x\mathop - \limits_{( + )} 12y = \mathop + \limits_{( - )} 57} \\
    \\
  22y = - 92 \\
 \]
Dividing by 22 we have
\[
   \Rightarrow y = - \dfrac{{92}}{{22}} \\
   \Rightarrow y = - \dfrac{{46}}{{11}} \\
 \]
We have found the value of \[y\] now we have to find the value of \[x\] . so we will substitute the value \[y\]to any one of the equation (1) or (2) . we will substitute the value of \[y\]to equation (1).
Therefore, we have \[3x + 2y = - 7\] \[ \Rightarrow 3x + 2\left( {\dfrac{{ - 46}}{{11}}} \right) = - 7\]
\[ \Rightarrow 3x - \dfrac{{92}}{{11}} = - 7\]
\[
   \Rightarrow 3x = - 7 + \dfrac{{92}}{{11}} \\
   \Rightarrow 3x = \dfrac{{ - 77 + 92}}{{11}} \\
   \Rightarrow 3x = \dfrac{{15}}{{11}} \\
 \]
Divide the above equation by 3 we have
\[ \Rightarrow x = \dfrac{5}{{11}}\]
Hence we got the unknown values \[x\] and \[y\]that is \[\dfrac{5}{{11}}\] and \[\dfrac{{ - 46}}{{11}}\] respectively,
We can check whether these values are correct or not by substituting the unknown values in the given equations and we have to prove L.H.S is equal to R.H.S
Now we will substitute the value of \[x\] and \[y\] in equation (1) so we have
\[3x + 2y = - 7\]
\[
   \Rightarrow 3\left( {\dfrac{5}{{11}}} \right) + 2\left( {\dfrac{{ - 46}}{{11}}} \right) = - 7 \\
   \Rightarrow \dfrac{{15}}{{11}} - \dfrac{{92}}{{11}} = - 7 \\
   \Rightarrow \dfrac{{15 - 92}}{{11}} = - 7 \\
   \Rightarrow \dfrac{{ - 77}}{{11}} = - 7 \\
   \Rightarrow - 7 = - 7 \\
 \]
Hence the values of the unknown that are \[x\] and \[y\] are the correct values which satisfy the equation.

Note: In this type of question while eliminating the term we must be aware of the sign where we change the sign by the alternate sign. In this we have a chance to verify our answers. In the elimination method we have made the one term have the same coefficient such that it will be easy to solve the equation.